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Unit 3 • Lesson 4 255
How much for just one?Oftenwemaywanttoknowthevalueofjustoneofsomething.Thevalue of one of something is called the unit rate.
We use this concept often without actually thinking about unit rate. For instance,inthegrocerystore,itiseasierifwecomparethecostofjustone unit, such as an ounce or a pound.
Ifweseeasignadvertisingfourboxesofcerealfor$10,wemightaskourselveshowmuchitcostsforjustonebox.
Let’s look at a problem involving unit rate.
Example 1
Find the unit rate.
Problem:
Itcosts$6tobuy3cartonsofmilk.Whatisthecostof1carton?
Set up a proportion with a variable.
CostCarton $6
3 = x1
Complete the proportion by finding the value of x.
CostCarton $6
3 = $21
One carton of milk costs $2. The unit rate is $2 per carton.
x is the cost for 1 carton.
Problem Solving: Solving Word Problems Using Unit Rates
Problem Solving: Solving Word Problems Using Unit Rates
Lesson 4
256 Unit 3 • Lesson 4
Lesson 4
Anotherwaywetalkaboutunitrateiswhenweusethetermmiles per hour.Thistermmeansthenumberofmileswetravelinonehour.Milesperhourisaunitrate.It’sthevalueofjustoneofsomethingeveryonehour. Example 2 shows this situation.
Example 2
Find the unit rate.
Problem:
Quentin’sdaddrovehimtoasoccertournament.Theydrove240milesin4hours.Abouthowfardidtheydriveinjust1hour?Whatwasthe rate in miles per hour?
Set up a proportion with a variable.
MilesHour 240
4 = m1
Complete the proportion by finding a value for m.MilesHour 240
4 = 601
We see now that m=60.ThismeansQuentinandhisdaddrove60milesin1hour.Anotherwayofsayingthisis“60milesperhour.”
The unit rate is 60 miles per hour.
Noticethateachproportionstartswiththeunitswritteninwords.
Writing the words out is a good habit to practice with this type of problem.Wewanttoremindourselveswhatthenumbersstandfor.Thatway, when we solve for a variable, we know exactly what that variable represents. In the example about Quentin and his dad, the variable m stands for miles. In the previous example, the variable x stands for the cost in dollars.
Example 3 shows another problem involving unit rate.
m is the miles traveled in 1 hour.
Unit 3 • Lesson 4 257
Lesson 4
Example 3
Find the unit rate.
Problem:
We are at the state fair. We use tickets to pay for the rides. Each ride requires the same number of tickets. We can take 4 rides on 20 tickets. How many tickets does it take for 1 ride?
Set up a proportion with a variable.
TicketsRide 20
4 = t1
Complete the proportion by finding the value for t.
TicketsRide 20
4 = 51
We need 5 tickets to take 1 ride.
t stands for the number of tickets.
We can see that t = 5.
258 Unit 3 • Lesson 4
Lesson 4
How do we solve word problems using unit rates?Sometimes we have to compute the unit rate for something to find the betterdeal.Weusuallyexpectthatitemsmarked“3for$ ” or“5for$ ”arebetterdealsthanbuyingjustoneitem.However, this is not always the case.
Tocomparethesedifferentpricingmethods,wefindtheunitrate.Example 1 shows how we analyze this type of situation to determine the better deal.
Example 1
Find the unit rate.
Problem:
Monicaneedssoup.Thestorehasaspecial—5cansfor$10.Ifshebuysjust1canofsoup,thecostis$2.20.Whichisthebetterdeal?
Wecomparethecostof1canofsoupfor$2.20tothespecialdealof5cansfor$10.Wesetupaproportiontofindtheunitrate.
CostNumber of Cans $10
5 = x1
• Firstweaskourselves,“1·?=5?”Theansweris5.
• Thenweneedtofindthevalueforx in the statement, “x ·5=$10.”
It’s$2.Wecompletetheproportionbyfillinginthevalueforx.
CostNumber of Cans $10
5 = $21
The unit price is $2.
Wenowcomparetheunitrateof$2tothecostfor1can—$2.20.Buyingsoupat5for$10isthebetterdealbecauseit’s$2percan,not$2.20percan.
Grocery stores often use this pricing method. We expect items marked “3for$ ” to be the best deals. However, that is not always the case.
Unit 3 • Lesson 4 259
Lesson 4
Example 2 shows a different situation. Sometimes the special pricing methods are not the best deal.
Example 2
Find the unit rate.
Problem:
Marcusneedstoweartiesforhisnewjob.Atthedepartmentstore,tiesare$19eachor3for$60.
Weneedtocomparethecostofjust1tietothespecialdealbyfindingthe unit rate. We set up the proportion like this:
CostNumber of Ties $60
3 = x1
• Firstweaskourselves,“1·?=3?”Theansweris3.
• Thenweneedtofindthevalueforx in this statement, “x ·3=$60.”
Thecostis$20.Wecompletetheproportionbyfillinginthevalueforx.
CostNumber of Ties $60
3 = $201
Whenwecomparethetwopricingmethods,weseethatthe“specialdeal” is not the better deal.
The unit rate is $20 per tie.
Ifwebuythetiesindividually,theyare$19pertie.
Problem-Solving ActivityTurntoInteractive Text, page 108.
Reinforce UnderstandingUse the mBook Study Guide to review lesson concepts.
260 Unit 3 • Lesson 4
Lesson 4
Activity 1
Simplify the ratios.
1. 714
12 2. 3
12 14 3. 4
20 15
4. 515
13 5. 8
48 16
Activity 2
Set up the unit rate problems as proportions. Tell what the variable represents. Show the units in words. Find the unit rate.
1. Nguyenpaid$40for4CDs.Ifall4costthesameamount,whatwasthecost ofjust1CD?
2. Sheldonpaid$180for6pairsofcontactlenses.Whatwasthecostofjust 1 pair of lenses?
3. Brittcando45sit-upsin3minutes.Howmanysit-upscanshedoinjust 1 minute?
Activity 3
Tell the better deal in each case by finding the unit rate.
1. What’sthebetterdeal,1applefor$.50,or3for$1?
2. What’sthebetterdeal,1pairofjeansfor$60or3for$200?
3. What’sthebetterdeal,1T-shirtfor$19or2for$40?
4. What’sthebetterdeal,1CDfor$15or2for$25?
5. What’sthebetterdeal,3juicesfor$1or$.75for1juiceinthevendingmachine?
Model 612 Answer: 6
12 = 12
Model Itcosts$8for4sandwiches.Howmuchdoesitcostforjust1sandwich?
Answer: CostSandwich $8
4 = x1 X stands for the cost of 1 sandwich.
CostSandwich $8
4 = $21 Thecostofonesandwichis$2.
Homework
CostCD
$404
$101
CostPair
$1806
$301
Sit upsMinutes
453
151
Copyright2010byCambiumLearningSoprisWest®.Allrightsreserved.Permissionisgrantedtoreproducethispageforstudentuse.
Unit 3 • Lesson 4 261
Lesson 4
Activity 4 • Distributed Practice
Solve.
1. 45 + 2
5 = x 15 2. 35 − w = 19
3. 4.75 + 2.98 = z 4. a + 385 = 410
5. 12 · 3
5 = b 3
10 6. 139.7 − 48.19 = c
7. 45 ÷ 1
5 = d 8. e ÷ 7 = 50
Homework
Copyright2010byCambiumLearningSoprisWest®.Allrightsreserved.Permissionisgrantedtoreproducethispageforstudentuse.